We propose two variants of one method. The first variant considers an exact application of the preconditioner in a BiCG setting for shifted systems, inspired by [1]}. The second variant incorporates an inexact application of the preconditioners in a flexible BiCG setting for shifted systems, based on the prior works [2]}, [3]}. We prove a bound on how inexactly we can apply the preconditioner \((K-\sigma M)^{-1}\)  without degrading convergence of the algorithm in general, inspired by work in [4]}. The first variant is appropriate only when an LU decomposition of a matrix of dimension \(n \times n\)  is feasible, where the inexact variant has the potential to solve a wider variety of large-scale problems. While the second variant is based on the first variant, the first variant may be useful in itself in cases where the exact LU decomposition is computable, as discussed in Section .
